E¼ 0 E¼ 1 E¼ 0 E¼ 1D¼ 0 þ (^1) 85 45 D¼ (^0) 45 30
D¼ 25060 D¼ 1 þ 29075
ORd¼ 2 : 27 ORd¼ 1 : 25
Statistical test of assumption:
Score test
Compares ordinal vs. polytomous
models
Test statisticw^2 underH 0
with df¼number of OR parameters
tested
Alternate models for ordinal data:
Continuation ratio
Partial proportional odds
Stereotype regression
III. Odds Ratios and
Confidence Limits
ORs: same method as SLR to
compute ORs.
Special case: one independent
variable
X 1 ¼1orX 1 ¼ 0
oddsðDgÞ¼
PðDgjX 1 Þ
PðD<gjX 1 Þ
¼expðagþb 1 X 1 Þ
The two collapsed tables are presented on the
left. The odds ratios are 2.27 and 1.25. In this
case, we would question whether the propor-
tional odds assumption is appropriate, since
one odds ratio is nearly twice the value of the
other.
There is also a statistical test – aScore test–
designed to evaluate whether a model con-
strained by the proportional odds assumption
(i.e., an ordinal model) is significantly different
from the corresponding model in which the
odds ratio parameters are not constrained by
the proportional odds assumption (i.e., a poly-
tomous model). The test statistic is distributed
approximately chi-square, with degrees of free-
dom equal to the number of odds ratio para-
meters being tested.
If the proportional odds assumption is inap-
propriate, there are other ordinal logistic mod-
els that may be used that make alternative
assumptions about the ordinal nature of the
outcome. Examples include a continuation
ratio model, a partial proportional odds model,
and stereotype regression models. These models
are beyond the scope of the current presenta-
tion. [See the review by Ananth and Kleinbaum
(1997)].
After the proportional odds model is fit and the
parameters estimated, the process for comput-
ing the odds ratio is the same as in standard
logistic regression (SLR).
We will first consider the special case where
the exposure is the only independent variable
and is coded 1 and 0. Recall that the odds
comparingDgvs.D<gis e to theagplus
b 1 timesX 1. To assess the effect of the exposure
on the outcome, we formulate the ratio of the
odds ofDgfor comparingX 1 ¼1 andX 1 ¼ 0
(i.e., the odds ratio forX 1 ¼1 vs.X 1 ¼0).
472 13. Ordinal Logistic Regression